where \(Z_{\rho }(t;x)\)  is the solution to (REF ) for a given \(\rho \) . Similar to the gradient flow for \(\Theta _{L,M}\)  and \(\Theta (t)\) , the probability distribution \(\rho \)  that encodes the configuration of \(\theta \)  in the mean-field setting, flows in the descending direction of \(E(\rho )\)  in pseudo-time \(s\) .
Considering that \(\rho (\theta ,t,s)\)  should be a probability density for all \(s\)  and \(t\) , this gradient flow is conducted in the Wasserstein metric. Following [1]}, [2]}, we have
\(\frac{\partial \rho (\theta ,t,s)}{\partial s}=\nabla _\theta \cdot \left(\rho (\theta ,t,s)\nabla _\theta \frac{\delta E(\rho )}{\delta \rho (s)}\right)\,,\quad s>0,\; t\in [0,1]\,,\) 
